Problem 11 - Calculus Early Transcendentals

Question

Sketches from properties Sketch a graph of a function that is continuous on \((-\infty, \infty)\) and has the following properties. Use a sign graph to summarize information about the function. $$\begin{aligned} &f^{\prime}(x)<0 \text { on }(-\infty, 2) ; f^{\prime}(x)>0 \text { on }(2,5) ; f^{\prime}(x)<0\\\ &\text { on }(5, \infty) \end{aligned}$$

Step-by-Step Solution

Answer

Question: Sketch a graph of a function that is continuous for all real numbers and has a first derivative that follows the given rules: - \(f'(x) < 0\) on \((-\infty, 2)\) - \(f'(x) > 0\) on \((2, 5)\) - \(f'(x) < 0\) on \((5, \infty)\) Answer: The graph of the function has a local minimum at \(x = 2\), a local maximum at \(x = 5\), and is decreasing on the intervals \((-\infty, 2)\) and \((5, \infty)\), while increasing on the interval \((2, 5)\).

Step 1: Understand the information given

Based on the information provided, we know that the function \(f(x)\) is continuous for all real numbers and its derivative follows specific rules for certain intervals: - \(f'(x) < 0\) on \((-\infty, 2)\) - \(f'(x) > 0\) on \((2, 5)\) - \(f'(x) < 0\) on \((5, \infty)\) We need to use this information to sketch the graph of the function.

Step 2: Identify the intervals where \(f(x)\) is increasing and decreasing

We can tell from the first derivative where the function is increasing and decreasing: - Since \(f'(x) < 0\) on \((-\infty, 2)\), the function is decreasing on this interval. - Since \(f'(x) > 0\) on \((2, 5)\), the function is increasing on this interval. - Since \(f'(x) < 0\) on \((5, \infty)\), the function is decreasing on this interval.

Step 3: Determine local extrema

Based on the increasing and decreasing intervals, we can identify the local extrema of the function: - Since the function is decreasing before \(x = 2\) and increasing after, there must be a local minimum at \(x = 2\). - Since the function is increasing before \(x = 5\) and decreasing after, there must be a local maximum at \(x = 5\).

Step 4: Create a sign graph

A sign graph can help us visualize the increasing and decreasing intervals of the function and find where the local extrema are. Here is the sign graph for the function \(f(x)\): \(x\) | \((-\infty, 2)\) | \(2\) | \((2, 5)\) | \(5\) | \((5, \infty)\) -- | -- | -- | -- | -- | -- \(f'(x)\) | - | | + | | - \(f(x)\) | Decreasing | Local Min | Increasing | Local Max | Decreasing

Step 5: Sketch the graph of the function

Using the information obtained from the intervals and the sign graph, we can now sketch a graph of the function \(f(x)\): 1. Start by drawing a local minimum at \(x = 2\). 2. Continue the graph to the left, making sure it is decreasing as \(x\) approaches \(-\infty\). 3. Draw a local maximum at \(x = 5\). 4. Continue the graph to the right, making sure it is decreasing as \(x\) approaches \(\infty\). Note that specifying the exact function is not required for this task, so focusing on the general shape that satisfies the given conditions is sufficient.

Key Concepts

First Derivative Test

The first derivative test is a critical tool used to analyze the behavior of functions. It helps determine where a function is increasing or decreasing and identify local extrema. By computing the first derivative of a function, which is the slope of the tangent line to the function at any point, you gain insight into the function's growth or decline rate.

For instance, if the first derivative, denoted as \(f'(x)\), is positive over an interval, it implies that the function is increasing there. Conversely, if \(f'(x)\) is negative, the function is decreasing. This test also assists in locating points where the derivative changes sign, which could indicate a local maximum or minimum—these points require further investigation to determine the exact nature of the extremum.

Increasing and Decreasing Intervals

Determining increasing and decreasing intervals of a function is fundamental to understanding its behavior. If the derivative of a function \(f'(x)\) is positive for all \(x\) in an interval, then the function is increasing on that interval. This means as \(x\) gets larger, so does \(f(x)\).

On the other side, if \(f'(x)\) is negative for all \(x\) within an interval, the function is decreasing, which means as \(x\) gets larger, \(f(x)\) gets smaller. A graph that portrays this behavior rises in increasing intervals and falls in decreasing intervals, adhering to the underlying algebraic signs of the first derivative.

Local Extremum

A local extremum refers to a point on the graph of a function where the function reaches a local maximum or minimum. At a local maximum, the function has higher values at that point than at nearby points; at a local minimum, the function has lower values. The first derivative test is often applied to identify potential local extrema by finding points where the derivative is zero \(f'(x) = 0\) or the derivative is undefined.

However, not every point where \(f'(x) = 0\) is a local extremum—it's only a candidate that must be verified. It's important to look at the behavior of \(f'(x)\) before and after these points. If the derivative changes sign, then it is indeed a local extremum; if it does not change sign, the function does not have a local extremum at that point.

Continuity of Functions

The continuity of functions is an essential concept in calculus, referring to functions that are unbroken and have no gaps, jumps, or points of discontinuity. For a function to be continuous at a particular point, the following must be true: it must be defined at that point, the limit as \(x\) approaches the point must exist, and the limit must equal the function value at the point.

The continuity of a function is a necessary condition for applying the first derivative and subsequently analyzing increasing and decreasing intervals and local extrema. It ensures smooth transitions without abrupt changes, allowing for accurate graph sketching and interpretation based on derivative information.

Sign Graph

A sign graph is a simple yet powerful visual tool that represents the signs of a function's derivatives and sheds light on the function's behavior. It often includes a number line indicating the intervals on which the function's first derivative \(f'(x)\) is positive or negative, and hence the intervals where the function is increasing or decreasing.

A typical sign graph will show critical points where \(f'(x) = 0\) and possible intervals of increase and decrease separated by these points. By placing '+' or '-' signs on the intervals, a sign graph provides a clear overview of the function's behavior, aiding in identifying local extrema and the function's overall trend.